TESLA COIL DESIGN FOR ELECTRON GUN APPLICATION M. Paralievξ, C. Gough, S. Ivkovic Paul Scherrer Institute, Accelerator Division 5232 Villigen PSI, Switzerland
describe our results of optimizing the parameters of the resonant air-core transformer.
Abstract The current project is to build an electron gun for X-ray Free Electron Laser (XFEL) application. The electron gun will utilize field emission and extreme accelerating gradient to achieve very low emittance. However for long-term study of cathode characteristics, a stable pulsed voltage in the megavolt range is needed. The first project phase is to design and test a 500kV pulser using a resonant air-core transformer (Tesla coil). Detailed results of simulations with Microwave Studio® and PSpice® for various coil geometries, tuning and coupling factors are given, and the optimum values for this application are given. In addition, experimental results are given for the most promising geometries.
II. RESONANT TRANSFORMER The electrical circuit of a lossless resonant pulse transformer is shown in Figure 1., where Sw is a switch, L and Ls are primary and secondary inductances, C and Cs are primary and secondary capacitances, and K is coupling factor. To derive the differential equations describing the electrical behavior of the circuit, the equivalent circuit shown in Figure 2. is used.
I. INTRODUCTION The requirement to sustain very high anode-cathode gradient (0.2…1GV/m) spoke in favor of having the shortest possible high voltage pulse. The cathode field emission current I is given by Fowler-Nordheim law shown below: ϕ 3/ 2 , (1) I = Ac1 F 2 exp − c2 F where A is emitting area, F is local field strength, ϕ work function of the emitting material, c1 and c2 are constants. Equation (1) shows that the emitted current is strongly dependant on the extracting field. In order to compare different field emitters, the pulser should have good stability and repeatability. The resonant air-core transformer technology was chosen because it is fast ( ≈ 200 ns), stable, fully linear and scalable. In the frequency domain, the resonant frequencies of the primary and secondary LC circuits separate with increased coupling. There is a particular coupling factor, namely 0.6, when one resonant frequency is exactly twice the other. This case is unique with its asymmetric voltage waveform and with its capability to reach the maximum voltage within one cycle (Figure 3.). This particular case of the resonant transformer we will call critically coupled. In the rest of this paper, we
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Figure 1. Simplified circuit of a resonant pulse transformer
Figure 2. Equivalent circuit of a resonant pulse transformer. All values are referred to the primary side. where i1 to i5 are the currents in branches with the shown directions, u1 and u2 are respectively the voltages in primary and secondary side and u3 is the voltage across L3 , which represents the coupling. All values are referred to the primary side. The values of the components in the equivalent circuit are the following:
C1 = C 2 = C ,
(2)
L1 = L2 = L(K + 1) ,
(3)
L3 = L
1− K 2 . K
(4)
Applying Kirchoff’s rule for junctions A and B we can write: i1 = i 3 + i 4 (5)
i5 = i 2 + i3
(6)
Substituting the currents and making some equivalent mathematical transformations, the following system of two second order differential equations can be obtained:
u1 + k1u1 − k 2 u 2 = 0 u2 + k1u 2 − k 2 u1 = 0
(7)
where U is the initial voltage of primary capacitor C. Going back to the original variables we find the solutions for primary voltage u1 and secondary one u 2 U u1 = [cos(ω 1 t ) + cos(ω 2 t )] , (21) 2 U u 2 = [cos(ω 1t ) − cos(ω 2 t )] . (22) 2 The solution consists of two cosine functions. In order to have an asymmetric output signal (Figure 3.) we are interested in the particular case when the second frequency ω 2 is exactly twice the first one ω1 . Using equations (17) and (18) the critical coupling K c can be calculated:
(8)
2ω1 = ω 2
where
k1 =
1 , LC (1 − K 2 )
(9)
k2 =
K . LC (1 − K 2 )
(10)
2
u1 − u 2 = z 2
(12)
As a result, two independent homogeneous second order differential equations are obtained:
z1 + (k1 − k 2 ) z1 = 0 , z2 + (k1 + k 2 ) z 2 = 0 .
(13) (14)
(24)
(25)
1.2
u1 u2
0.8
Normalized Amplitude
(11)
1 LC (1 − K c )
K c = 0 .6
By adding and subtracting equations (7) and (8), and using substitutions (11) and (12), the equations in the system can be decoupled [1].
u1 + u 2 = z1
1 = LC (1 + K c )
(23)
0.4 0 -0.4 -0.8 -1.2 0
1
2
3
Cycles
Figure 3. Normalized primary (u1) and secondary (u2) voltages
The solutions of (13) and (14) are shown below:
z1 = A cos(ω1t +ϕ 1 ) ,
z 2 = B cos(ω 2 t +ϕ 2 ) ,
(15) (16)
where A, B, ϕ1 and ϕ 2 are integration constants and
ω1 =
1 , LC (1 + K )
(17)
ω2 =
1 . LC (1 − K )
(18)
Using the initial conditions the final solutions for z1 and z 2 can be found:
z1 = U cos(ω1t ) ,
z 2 = U cos(ω 2 t ) ,
(19) (20)
III. PARAMETRIC STUDY In order to be able to evaluate the behavior of the transformer, two criteria have been chosen to be maximized: output peak voltage and negative to positive amplitude ratio. The equivalent circuit was simulated with PSpice®. All values are normalized to the critical coupling case ( K = 0.6 ), and primary and secondary sides tuned to one and the same frequency. Two cases were covered to check the influence of damping on the circuit behavior. For the first case the damping factor was small (
